We first prove the following standard counting result on the size of the sets \(F(\alpha ; t,\epsilon )\) .
This is essentially the same as, for example, [1]}.

Let \(\alpha \ge 0\)  be arbitrary and \(q\in \partial \tau _{\mathcal {L}}^*(\alpha )\) .
Then there exists some \(r>0\)  such that for all \(0<t<r\) ,
\(\#F(\alpha ;t,\epsilon )\le t^{-\tau _{\mathcal {L}}^*(\alpha )-(1+|q|)\epsilon }.\) 
